The Cancellation Law for Groups

The Cancellation Law for Groups

On the Basic Theorems Regarding Groups page we looked at a whole bunch of theorems. We will now look at another important property of groups in that the cancellation law always applies.

Theorem 1: Let $(G, *)$ be a group and let $a, b, c \in G$. If $a * b = a * c$ or $b * a = c * a$ then $b = c$.
  • Proof: Let $a^{-1} \in G$ denote the inverse of $a$ under $*$. Suppose that $a*b = a* c$. Then:
(1)
\begin{align} \quad a * b = a * c \\ \quad (a^{-1} * a) * b = (a^{-1} * a) * c \\ \quad e * b= e * c \\ \quad b = c \end{align}
  • Similarly, suppose now that $b*a = c*a$. Then:
(2)
\begin{align} \quad b * a = c * a \\ \quad (b * a) * a^{-1} = (c * a) * a^{-1} \\ \quad b * (a * a^{-1}) = c * (a * a^{-1}) \\ \quad b * e = c * e \\ \quad b = c \quad \blacksquare \end{align}

It is very important to note that the cancellation law holds with regards to the operation $*$ for any group $(G, *)$. We will see that the cancellation law does not necessarily hold with respect to an operation on a set when we look at algebraic structures with two defined operations.

It is also important to note that if $a * b = c * a$ or $b * a = a * c$ then we cannot necessarily deduce that $b = c$ because we would then require the additional property that $*$ is commutative which is not one of the group axioms (but instead one of the Abelian group axioms).

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